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EquivalenceclassesofFibonaccilatticesandtheirsimilarityproperties N. Lo Gullo,1,2 L. Vittadello,1 M. Bazzan,1,2 and L. Dell’Anna1,2 1Dipartimento di Fisica e Astronomia, Universita` degli Studi di Padova, Padova, Italy 2CNISM, sezione di Padova (Dated:July28,2016) Weinvestigate,theoreticallyandexperimentally,thepropertiesofFibonaccilatticeswitharbitraryspacings. Differently from periodic structures, the reciprocal lattice and the dynamical properties of Fibonacci lattices dependstronglyonthelenghtsoftheirlatticeparameters,evenifthesequenceoflongandshortsegment,the Fibonaccistring,isthesame. Inthisworkweshowthat,byexploitingaself-similaritypropertyofFibonacci stringsunderasuitablecompositionrule,itispossibletodefineequivalenceclassesofFibonaccilattices. We showthatthediffractionpatternsgeneratedbyFibonaccilatticesbelongingtothesameequivalenceclasscan 6 berescaledtoacommonpatternofstrongdiffractionpeaksthusgivingtothisclassificationaprecisemeaning. 1 Furthermoreweshowthat,throughthegaplabelingtheorem,gapsintheenergyspectraofFibonaccicrystals 0 belongingtothesameclasscanbelabeledbythesamemomenta(uptoaproperrescaling)andthatthelarger 2 gapscorrespondtothestrongpeaksofthediffractionspectra. Thisobservationmakesthedefinitionofequiv- l alence classes meaningful also for the spectral, and therefore dynamical and thermodynamical properties of u quasicrystals. Ourresultsapplytothemoregeneralclassofquasiperiodiclatticesforwhichsimilarityundera J suitabledeflationruleisinorder. 6 2 PACSnumbers: ] r e I. INTRODUCTION II. GENERALIZEDFIBONACCILATTICES h t o Inonedimension(1D)theparadigmofaquasicrystalisthe t. Sincethefirstexperimentalproofoftheexistenceofsolids Fibonaccilattice(FL).TheFLisa1Dlatticewhoseadjacent a pointshavedistancesbelongingtotheset{L,S},standingfor lacking of translational invariance, but exhibiting a discrete m LongandShortrespectively,whicharearrangedaccordingto Braggdiffractionspectrum[1],thestudyofquasicrystalsat- - tractedquitealotofattention.Theimpactofthisdiscoveryon agivensequence. Suchalatticecanbeconstructedbymeans d of the cut and project technique [3, 4, 21] thus obtaining for n thescientificcommunitywassuchthatin1992theformerdef- thecoordinatesofpointsontherealline[2](inunitsofS): o initionofcrystalhadtobemodifiedinordertoincludethose c structures whose diffraction pattern witnesses long range or- 1 (cid:106)n(cid:107) [ der yet lacking translational invariance [2–4]. More gener- xη =n−1+ (1) n η τ 2 ally, the study of quasiperiodic geometries has been recently v thesubjectofdifferentfieldsalldevotedtothepropagationof where n is a natural positive number,(cid:98)x(cid:99) is the integer part 9 wavesthroughquasiperiodicpotentials. Thespectralproper- of x and η = S/(L − S). The most common instance 9 tiesofquasicrystalshavebeenrecentlyusedtoengineertopo- (cid:0)√ (cid:1) found in literature is obtained for η = τ = 5+1 /2, 4 logicalpumpinginopticalwaveguides[5–7]andinultracold 7 the golden ratio. In this case the canonical FL (CFL) is ob- gases[8,9].Engeneeringofquasiperiodicstructureshavealso 0 tained, such that the lengths are (up to a simple rescaling): been employed in optical dielectric multilayers for resonant . L = 1 + 1/τ = τ, and S = 1. Nevertheless it is possi- 1 transmission [10], solar energy harvesting [11], plasmonics ble to construct Fibonacci lattices with η (cid:54)= τ (see App. A). 0 6 [12,13]andnonlinearoptics[14,15]. The distances ∆n = xηn+1 −xηn are either L = 1+1/η or 1 S =1andtheyarearrangedaccordingtotheFibonaccistring : Dynamical and transport phenomena in this kind of struc- (FS)LSLLSLSLLSLLSLSLLSLSL···.Thelatterisany v tures are also radically different compared to periodic me- wordmadeoftwoletters,LandS,obtainedbymeansofthe i X dia[16–20]. Forusualperiodicarrangements,dynamicaland substitutionruleS → LandL → LS startingfromtheletter r thermodynamicalpropertiesaredirectlyrelated,viatheBloch L. WenoticethataFSitselfisindependentontheparameter a theorem, to the geometry of the system. Quasiperiodic ge- ηanditonlydependsonthefactor1/τ. ometries,instead,lacksoftranslationalinvariancesothatadi- Conversely,givenaninfiniteFS,acompositionrule(LS → rectrelationbetweentheirstructureandtheirdynamicalprop- L(cid:48) and L → S(cid:48)) can be defined such that the old and the erties is not generally known. It would be therefore very in- newstringsarethesameduetothepeculiarpropertiesofthe terestingtofindasortofclassificationenablingonetogroup Fibonacci strings, as shown in Fig. 1. For the special case togetherdifferentaperiodicsystemsonthebasisofsomesim- η = τk, with k a non-vanishing integer i.e. for the canoni- ilaritybetweentheirgeometricarrangements.Inthispaperwe cal FL, this leads to a peculiar property: the new FL can be attempttodefinesuchaclassification,showingthatquasiperi- rescaledtotheoriginalone. Thiscaseisthemostcommonly odicstructureswhosegeometryisrelatedbyasuitablemath- encountered in literature, accompanied by the statement that ematicaltransformationsharethemaincharacteristicsoftheir the CFL is self-similar. It should be stressed however that reciprocallatticeandoftheirpseudo-bandstructure. this is not true for the general case η (cid:54)= τk. In this case, a 2 for irrational η we can resort to its continued fraction repre- L S L L S L S L L S L sentationinordertosetthewantedprecisiontotheabovecon- dition. L' S' L' L' S' L' S' LetusnowconsidertwoFLsbelongingtothesameequiv- alenceclassxηn0 andxηn1,withη1 =1+1/η0. Bydefininghn Figure1: Thecompositionrules(LS → L(cid:48), L → S(cid:48))onasemi- (kn)andh(cid:48)n(kn(cid:48))asthenumeratoranddenominatorofthen- infinite Fibonacci lattice reproduces another Fibonacci lattice with thrationalapproximantsof1+1/η (1+1/η ),thefollowing 0 1 differentlatticeparameters. relationshold: k = h +h(cid:48) andk(cid:48) = h . Thepositionof n n n n n thebrightestpeaksoftheFLxη1 arethenrelatedtothoseof n theFLxη0 by: n non-canonical Fibonacci lattice and the one obtained by ap- plyingthecompositionrulearecharacterizedbytwodifferent Q1(kn,kn(cid:48))=η1Q0(hn,h(cid:48)n) (4) lenghtratiosη andη becauseL(cid:48)/L (cid:54)= S(cid:48)/S. Thereforethe 1 2 In other word, althought the two Fibonacci lattices xη0 and newlatticecannotbetransformedintotheoldonebyasimple n rescaling. C(xηn0) = η1xηn1 cannot be rescaled one over the other (for the general case η (cid:54)= τ), their brightest peak pattern can, as We call C the operator corresponding to the effect of the aconsequenceofthefactthattheyarerelatedbythecompo- compositionruleontheFLxη. Itisnotdifficulttoshowthat n sition rule. Also the intensities of the brightest peaks can be C(xηn) = η1xηn1, whereη1 = 1+1/η: thecompositionrule relatedasI(η q,η )≈I(q,η )+ 1(1−I(q,η ))(forqsuch maps a FL xη into another FL, characterized by a new ratio 1 1 0 τ 0 n thatI(q,η ) > 0.5)showingthatthepeaksofthescaledlat- η and rescaled by η (see Fig. 1). If η = τ then η = τ 0 1 1 1 tice are even brighter than those of the original lattice. This (recall that τ2 −τ −1 = 0) and therefore the CFL is self- drives to the important conclusion that FL belonging to the similar. Ifthecompositionruleisappliedk times, theinitial sameequivalenceclasshavediffractionspectracharacterized FLismappedinto bythesamepatternofbrightestpeaks,andare,inthissense, (cid:32) k (cid:33) (cid:18) (cid:19) similar. C(k)(xηn)= (cid:89)ηi xηnk = Fk+1+ Fηk xηnk (2) i=1 III. SIMILARITYOFDIFFRACTIONPATTERNS where η = 1+1/η (η = η) and F are the Fibonacci i i−1 0 k numbers. Withthehelpofthisconceptwedefineequivalence Toquantifythedegreeofsimilaritybetweenthetwospec- classes for FLs by means of the following equivalence rela- tra, we use the Kullback-Leibler divergence (KLD), a quan- tion: tity useful to compare two distributions (normalized to unity over a common support). Let us consider the diffraction Definition - Two Fibonacci lattices xηna and xηnb are equiva- spectra I(q,ηα) and I(q,ηβ) of two arbitrary FL’s charac- lent(xηna ∼ xηnb)iftheyarelinked,uptoaproperrescaling, terized by ηα (cid:54)= ηβ. We define the normalized spectrum: bymeansofthecompositionruleC. Thelatticewiththemini- P(νq,η) = I(νq,η)/(cid:82)∞dkI(νk,η) where we introduced 0 mumη0suchthat1/(η0−1)isfiniteandpositiveiscalledthe ascalingparameterν. generatoroftheequivalenceclasswhichisdenotedby[η0]. TheKLDisdefinedas: Asimplewayoflabelingtheelementsofagivenequivalence D(η ,η ,ν)=(cid:90) ∞dkP(k,η )log(cid:18) P(k,ηα) (cid:19). (5) classisbymeansofthecontinuedfractionrepresentationfor α β α P (νk,η ) 0 β the{η }. i Bydefinitiononehasthatthemoresimilarthetwodiffraction Because of this, for quasiperiodic structures it is of lim- spectra, the smaller the value of the KLD. We will use it to ited practical utility to talk about the support of the diffrac- measureif,forgivenη andη ,thereexistascalingparameter tion pattern. It is more meaningful to describe the diffrac- α β ν forwhichthetwospectralooksimilar. tionspectrum(andthereciprocallattice)intermsofthepeaks In Fig. 2 a) and b) we plot 1/D(ηα,η,ν) comparing two whicharesignificantlyclosetoone,whichwillbereferredto 0 generators corresponding to ηa = 6/11 and ηb = 1/6 with as brightest peaks. By means of the cut and project method 0 0 FLs obtained from them by applying the composition rule outlined in App. A it is possible to show (see App. B) that the intensity I(q,η) at points q = Q(h,h(cid:48)) is given C(nα) respectively na = 1 and nb = 3 times. The inten- by sinc2(Q (h,h(cid:48))∆), where Q (h,h(cid:48)) = 2πd−1(h(1 + sities I(q,η) are evaluated by means of eq.B15 for lattices ⊥ ⊥ with N = 300 points. The maxima (minima of D) in the 1/η)−1−h(cid:48))and∆=τ(η/(η+1))/2. Thereforethebright- two figures correspond to (η,ν) = (ηa,ηa) and (η,ν) = estpeaksarefoundforpairs(h,h(cid:48))suchthatQ (h,h(cid:48)) ≈ 0 1 1 ⊥ (ηb,ηbηbηb) respectively, in agreement with eq. (4). This andthusfor 1 1 2 3 shows that two FL’s produce a similar diffraction pattern if h 1 andonlyiftheycanberelatedviaEq. (2)andthereforeonly =1+ . (3) h(cid:48) η iftheybelongtothesameequivalenceclass. In order to test our results on a real case, we performed a Sincehandh(cid:48) areintegers,theaboveconditioncanbesatis- diffraction experiment on two quasiperiodic diffraction grat- fiedexactlyonlyifη isarationalnumber. Ontheotherhand ings prepared using a photorefractive direct laser writing 3 D Figure 3: (Color online). a) Comparison between the theoretical diffractionpatternfortheFLxη0a (ηa = 0.5454, solidbluecurve) andtheexperimentaloneproducnedby0aFLC(cid:16)xη0a(cid:17)(solidredcurve, n N = 300 lines and L = 23 µm and S = 17 µm, ηa = 2.83). 1 ThetheoreticalspectrumhasbeenrescaledinaccordancewithEq. (4) and corrected for the structure factor contribution. b) The in- verse of KL divergence (D−1(ηa,η,ν)) between the (normalized Figure 2: (Color online) Inverse of KL divergence 1/D(ηa,η,ν) 1 0 over the interval) experimental diffraction pattern and the theoreti- comparing the diffraction spectra of the generator of a given class caldiffractionpatternsfordifferentgeneratorsanddifferentscaling. xηn0 with another Fibonacci lattice, for two choices of η0. a)η0a = Here∆ηa =η−ηaand∆ν =ν−1/ηa. Themaximum(minima 6/11;b)ηb =1/6.Here∆ηa =η−ηa,∆νa =ν−ηaand∆ηb = 0 0 0 1 1 ofD)isat(η,ν)=(ηa,1/ηa). η−ηb,∆νb =ν−ηbηbηb.Themaxima(minimaofD(η ,η ,ν)) 0 0 1 1 2 3 α β areobtainedat∆ηa,b = 0and∆νa,b = 0,indicatingthatthetwo spectra with the higest degree of similarity corresponds to lattices ηaxη1a =C(cid:16)xη0a(cid:17)andηbηbηbxη3b =C(3)(xη0b)respectively,i.e.the arangeofη0 andscalingfactorν. InFig. 3b)weshowitex- 1 n n 1 2 3 n n plicitlyforthegratingwithηaanditisclearthatthemaximum 1 firstandthethirdelementoftherespectiveequivalenceclasses. c) ofD−1 (minimumofD)isfoundatη = ηα andν = η−1. DirectcomparisonofthetwodiffractionspectrafortwoFLsxηn0aand Similarresultsareobtainedforthegrat0ingηb0. 1 ηaxη1a.ThemostprominentpeaksofthediffractionpatternI(q,ηa) 1 n 0 correspondwiththoseofthe(rescaled)spectrumI(ηaq,ηa). 1 1 IV. ENERGYSPECTRACOMPARISON We have therefore shown that all the FL belonging to the (DLW) technique [25, 26]. We used three gratings made up sameequivalenceclasshavediffractionspectrathat,although ofN =300linesallwritteninthesamesubstrate: (a)aperi- not equals, are characterized by a similar pattern of bright odicgratingwithspacingL=23µm;twoFibonaccigratings with (b) L = 23µm and S = 17µm (ηa = 17/6) and (c) peaks. Thisfindingisofcrucialimportancenotonlyinscat- L = 23µmandS = 15µm(ηb = 15/8)1respectively. Sofar tering phenomena but also in transport ones. In fact in a re- 3 cent work [23] a method to unambiguously link the gaps in weconsideredpointlattices,butrealstructuresareconstitued the integrated density of states to the brightest peaks in the bysomephysicalentity(basis)arrangedonthepointsofour diffraction pattern of the underlying potential has been pro- quasi-periodic Fibonacci lattice. (For a detailed description posed.Thisismoregeneralhasithasbeenshowninaseminal of the experimental set up see App. E). For these cases, the paperbyLuck[22]. LetusconsiderforexampletheHamilto- diffraction pattern is given by the sum in Eq. (B15) multi- nianforaparticleina1Dlattice: plied by the square modulus of a structure factor. The latter, in general, does not posses any scaling property and there- (cid:126)2 d2 foreitisnecessarytocorrectforitwhencomparingdifferent Hˆ = − +V(x) (6) 2 dx2 lattices. We did this experimentally by using the data of the (cid:90) periodicgratingtoextractaphenomenologicalexpressionfor V(x) = −V dyf(x−y)(cid:88)δ(y−x ) (7) 0 n the structure factor as a function of q. In Fig. 3 a) we com- n pare the experimental data relative to the grating ηb with the 1 theoretical diffraction pattern obtained from the generator of where x are the local minima of the potential and f(x) is n the corresponding equivalence class, ηb = 1/6. We observe introduced to account for the detailed shape of the potential 0 that,oncethespectrahavebeenrescaledinqfollowingeq.(4) minima (V > 0). We will consider the case x = xη ac- 0 n n andcorrectedinordertotakeintoaccountthestructurefactor cordingtothequasi-periodicsequenceofEq.1. InRef.[23] contributiontotheintensityofthepeaks,themostprominent it has been shown that it is possible to label the energy gaps diffraction features of the generator can be found in the ex- bymeansofthebrightestpeaksofthediffractionspectrum.In perimentaldataatthecorrectq positions. Thedegreeofsim- particularonehastoconsiderthepseudo-momentaqatwhich ilaritybetweenthespectrumofthegeneratorandtheexperi- the square of the Fourier transform of V(x) acquires values mentaloneisconfirmedbytheKLdivergenceDbetweenthe greater than a given threshold. This effectively corresponds experimentaldatapointsandthetheoreticaldiffractionspec- tochoosewhichfreestatesareeffectivelycoupledbythepo- trum(withtheinclusionofthestructurefactor)calculatedfor tential and, therefore, where wider gaps open in the single 4 shownthatitispossibletogroupdifferentFibonaccilattices intoequivalenceclasseswhoseelementssharethemainstruc- tural and dynamical properties as witnessed by their diffrac- tionspectraandtheenergygaps. Theseresultsshowthatthe concept of equivalence classes for FLs has not only a geo- metrical meaning but also an important role in the scatter- I ing,dynamicalandthermodynamicalpropertiesofthesystem, contained in the energy spectrum. It is worth stressing once again that this is a consequence of the self-similarity of FSs 0. under the composition rule and that FLs belonging to differ- entequivalenceclassescannotberescaledoneovertheother. Thegeneratorofaclassis,inthissense,thesimpleststructure giving a diffraction pattern which contains the main features common to all of the other elements of the class. Although Figure 4: (Color online). Plot of the energy level spacings of the wefocusedontheFibonaccilattices,ourargumentsapplyto Hamiltonian in Eq.(7) for a potential V(x) having minima at the themoregeneralclassofquasicrystalsforwhichdeflationor points of FLs xη with (blue dots) η = ηa, (red crosses) ηa. Be- lowwecomparenthegapswiththebrightest0peaksofI(q/ηa,1ηa). inflationrulescanmaptheinitiallatticesintoasimilarones. 1 1 particlespectrum. OneoftheresultspresentedinRef.[23]is Acknowledgments thatthisisequivalenttosetathresholdtotheintensityofthe peaks in the Bragg spectrum of the lattice. This can be eas- NL and LD acknowledge financial support from MIUR, ilyseenbyconsideringtheFouriertransformofthepotential throughFIRBProjectNo. RBFR12NLNA 002. LVandMB V(x),namelyV(q) = (cid:82) eıxqV(x)dxwhosesquaremodulus acknowledge financial support from Universit degli studi di isgivenby: Padova through Chip & CIOP project No. CPDA120359. The authors are thankful to Prof. Camilla Ferrante and Dr. |V(q)|2 =S(q)I(q,η) (8) Nicola Rossetto, from the Dipartimento di Chimica, Univer- whereS(q)isthesquareoftheFouriertransformoff(x)and sit di Padova, for providing access to the direct laser writing I(q,η)isgivenbyEq.B15.Itisclearfromwhatshownabove setup. The authors thank J. Settino for providing the energy andconfermedbytheexperimentonthediffractionpatterns, spectraofaparticleinaFibonaccilikelattice. that,apartfromthecontributionoftheactualformofthepo- tential(whichplaysaroleanalogoustothestructurefactorin diffractionexperiments),theenergypseudo-bandstructurein AppendixA:GeneralizedFibonaccilatticesfromcutand reciprocalspacehasthesameshape(uptoarescaling)forthe projectmethod lattices beloging to a given class. As an example, we com- puted the spectra of Eq. (7) in the case of Gaussian wells, The Fibonacci lattices we considered in the main text can namely f(x) = e−x2/2σ2 for a system with N = 100 min- beconstructedbymeansofthecutandprojecttechnique.One ima and x = xη with η = ηa,ηa and η = ηb,ηb. We possibleconstructionhasbeenpresentedinref.[21]Weprefer n n 0 1 0 3 choose V = 12, σ = 0.1. In Fig. 4 we plot the energy toresorttoamorestandardoneandinwhatfollowswewill 0 level spacing for lattices characterized by ηa = 6/11 (blue generalizetheonegiveninref. [3]. dots) and ηa = 17/6 (red crosses) both b0elonging to the Letusintroduceatwo-dimensionalperiodiclatticeIp and 0 2 equivalence class [η0a]. The momenta on the x-axis serve as its lattice vectors e1 and e2 such that any point of the lat- areferencewithrespecttothefreeparticledispersionrelation tice can be written as p = n1e1 + n2e2 with n1,n2 ∈ Z. ((cid:15)k = k2/2,V(x) = 0) to show where the potential V(x) Furthermore we introduce the line lτ whose unit vector is opens the gaps. After rescaling the momenta for the lattices ˆlτ = (cos(θτ),sin(θτ)) and the unit vector orthogonal to it withη = ηa byν = ηa wecanclearlyseethatthegapsap- ˆl⊥ = (sin(θ ),−cos(θ )) such that tan(θ ) = τ−1 where 1 a 1 τ √τ τ τ pearatthesamepoints. Ontheotherhandthesepointscorre- τ = (1 + 5)/2. The canonical Fibonacci lattice is con- spondtothebrightestpeaks,whereI(q/ηa,ηa)calculatedby structed by projecting on the line l the points of a square 1 1 τ Eq.(B15)issizeable.Similarresultsareobtained(notshown) lattice(e ·e = 0,|e | = |e |)thepointswhoseVonro¨ıcell 1 2 1 2 for the equivalence class [ηb] with ηb = 1/6 by considering iscutbythelineitself. Letusnoticethatwiththisprocedure 0 0 thetwolatticescharacterizedbyηb andηb =15/8,underthe thedifferentpointsareunambiguouslynumberedontheline 0 3 scalingνb =η1bη2bη3b. lτ onceanoriginandadirectionhavebeenchosen.Wearego- ingtoconstructourFibonaccilatticesusingthisdefinitionbut allowingthetwo-dimensionallatticetobegenericasinFig.5. V. CONCLUSIONS Nevertheless we will see that in order to obtain a Fibonacci lattice, namely a one-dimensional set of points whose dis- In conclusion, we investigated the diffraction spectra of tances are distributed according to the Fibonacci strings and FL’s in the general case η = S/(L−S) (cid:54)= τ. We have with the wanted ratio between long and short segments, we 5 willhavetorestrictthesetoftheallowedtwodimensionallat- tices. Followingthediscussioninref.[3]inorderfortheline γˆl tocuttheVonro¨ıcellcenteredatpointpithastointersect τ the secondary diagonal of the cell (joining the northwest to southeastpointofthecell). Thediagonalslieonlinesparallel toδ(e −e )andwhosepointsaregivenbyδ(e −e )+ne 1 2 1 2 1 with m ∈ Z. Their intersection with the line γˆl occurs at √ τ points(cid:0)n, n(cid:1)awherea=(τ2/ 1+τ2)A /(e −e )·ˆl⊥ τ τ2 u.c. 1 2 τ andA =(e ∧e )·zˆisthe(oriented)areaoftheunitcell u.c. 1 2 ofthelattice. Figure 5: (Color online) Construction of a generalized Fibonacci lattice from a 2D periodic lattice by means of the cut and project method.Bluedotsaretheprojectionofpointsofthe2Dlatticewhose ThisintersectionpointsareinsidetheVonro¨ıcellcentered Vonro¨icellsarecutbythelinelτ. atpoint(u,v)ifandonlyif |(e −e )·xˆ| |(e −e )·xˆ| u− 1 2 < naτ−1 <u+ 1 2 (A1) 2 2 |(e −e )·yˆ| |(e −e )·yˆ| v− 1 2 < naτ−2 <v+ 1 2 (A2) 2 2 On the other hand each point of the lattice can be written as projected.Thuswecanwriteu=n (e −e )·xˆ+ne ·xˆand 1 1 2 2 n e +n e andn +n =nbecauseitisthen-thpointtobe v =n (e −e )·yˆ+ne ·yˆandaboveinequalitiesbecome 1 1 2 2 1 2 1 1 2 2 (cid:16) s (cid:17) (cid:16) s (cid:17) n − x (e −e )·xˆ< n(aτ−1−e ·xˆ) < n + x (e −e )·xˆ (A3) 1 2 1 2 2 1 2 1 2 (cid:16) s (cid:17) (cid:16) s (cid:17) n − y (e −e )·yˆ< n(aτ−2−e ·yˆ) < n + y (e −e )·yˆ (A4) 1 2 1 2 2 1 2 1 2 wheres =Sign((e −e )·xˆ)andsimilarlyfors .Bymeans coordinatesonthethelinel : x 1 2 y τ of the expression for a it is easy to prove that (a τ−1 −e · 2 xˆ)/(e −e )·xˆ = (a τ−2−e ·yˆ)/(e −e )·yˆandthus 1 2 2 1 2 (cid:22) (cid:23) thetwoinequalitiesareequivalenttotheinequality: n x(cid:48) =ne ·ˆl +(e −e )·ˆl (A7) n 2 τ 1 2 τ β (cid:18) (cid:19) (cid:18) (cid:19) 1 n 1 n1− 2 < β < n1+ 2 (A5) Bynormalizingwithrespecttoe2·ˆlτ weeventuallyobtainthe one-dimensionallatticeofpoints e ·ˆl⊥ 1 β =1− 1 τ =1+r (A6) e ·ˆl⊥ τsin(α)−cos(α) 2 τ (cid:22) (cid:23) 1 n x =n+ (A8) where cos(α) = e1 ·e2/(|e1||e2|) and r = |e1|/|e2|. Be- n η β iinngeqnua1liatniesinttoegbeersnautimsfibeedritshethoatnlny1p=oss(cid:98)inβbi(cid:99)liwtyhfeorer t(cid:98)hxe(cid:99)aibsothvee η−1 =(cid:32)e1·ˆlτ −1(cid:33). (A9) integerpartofx. Afterprojectingontol ,then-thpointhas e ·ˆl τ 2 τ 6 In order for the above to be a Fibonacci lattice we require |(e − e ) · ˆl⊥| around the line γˆl . It is easy to see that 1 2 τ τ β = τ which is the case for τr = (τsin(α)−cos(α)) and A(q ) = A (q ,0). We therefore turn to the calculation of (cid:107) X (cid:107) thusη =(τ +tan(α))/((τ −1)tan(α)−τ2). Moreoverwe thelatter. Byintroducingthemassdensityofthetwodimen- (cid:80) have to require that r > 0 and η > 0 which is the case for sional lattice ρ((cid:126)r) = δ((cid:126)r −(cid:126)r ) and its Fourier tan−1(2τ +1) < α < tan−1(−τ)+π. As it can be seen transformρ((cid:126)r)=(cid:82) dk dmk1me−2ı(cid:126)r·(cid:126)kρ˜((cid:126)k)mw1emc2anwrite: ⊥ (cid:107) fromfigs.6foranygivenη >0therecorrespondapair(r,α): 1 1 (cid:90) A (q ,q ) = lim dk dk (B3) ητ +1 X (cid:107) ⊥ L→∞L2∆ ⊥ (cid:107) tan(α) = τ2 (A10) (cid:90) ∆ (cid:90) ∞ η−τ dx dx eı(cid:126)r·(q(cid:126)−(cid:126)k) ρ˜((cid:126)k) ⊥ (cid:107) (τtan(α)−1) −∆ −∞ r = (A11) (cid:112) τ 1+tan2(α) Theaboveintegralscanbecalculated: andthereforeatwodimensionallatticewhoseprojectionon A(q )=A (q ,0)=(cid:90) dk sin(k⊥∆)(cid:90) dk ρ˜((cid:126)k)δ(k −q ) thelinelτ returnsthewantedFL: (cid:107) X (cid:107) ⊥ k⊥∆ (cid:107) (cid:107) (cid:107) (B4) 1 (cid:106)n(cid:107) xη =n−1+ (A12) which expresses the fact that the diffraction pattern of an in- n η τ finite projected quasicrystal is the convolution of the Dirac (cid:32) (cid:33) e ·ˆl combformedbytheperiodichigherdimensionalperiodiclat- η−1 = 1 τ −1 , (A13) ticewiththesincfunctionintheorthogonalspace. e ·ˆl 2 τ Inparticularρ˜((cid:126)k)isaDiraccombpeakedatpointsk = hh(cid:48) whereweshiftedthewholelatticeinorderforthefirstpoint hw1+h(cid:48)w2whereweintroducedthereciprocallatticevectors tohavecoordinatex =0onthelinel . forthedualofthetwo-dimensionalperiodiclatticeIp: 1 τ 2 2π w = (e −e ·eˆ eˆ ) (B5) 1 l 1 1 2 2 1 2π w = (e −e ·eˆ eˆ ) (B6) 2 l 2 2 1 1 2 (B7) whereeˆ =e /|e |andl =|e |2−|e ·eˆ |2andsimilarlyfor i i i 1 1 1 2 l . Itiseasytocheckthatw ·e = 2πδ . Inwhatfollows 2 i j ij Figure6: (Coloronline)a)Valuesofr =|e1|/|e2|asafunctionof we assume that units are scaled such that e2 · ˆlτ = 1. In η. The red dot corresponds to the point (η,r) = (τ,1) for which ordertoevaluatetheparallelandperpendicularcomponentsof thecanonicalFibonaccilatticeisobtained. b)Valuesoftheangleα vectorsbelongingtothereciprocalspaceweneedtoevaluate betweene1ande2asafunctionofη.Thereddotcorrespondstothe wi·ˆlτ andwi·ˆlτ⊥. Inordertodosoitisusefultorewritethe point(η,α) =(τ,π/2)forwhichthecanonicalFibonaccilatticeis vectorse aslinearcombinationsofˆl andˆl⊥bymeansofthe obtained. i τ τ expressionsforη,βandtherelationbetweentanαandη.We thusobtain: (cid:18) (cid:19) (cid:18) (cid:19) AppendixB:Diffractionpattern e = 1+ 1 ˆl + 1 1+ 1 ˆl⊥ (B8) 1 η τ τ η τ Weareinterestedinthecalculationofthequantity: (cid:18) 1(cid:19) e = ˆl − 1+ ˆl⊥ (B9) 2 τ η τ A(q(cid:107))=Nl→im∞N1 (cid:88)eıxηnq(cid:107) (B1) Itisnoweasytocheckthat: n 2π 2π w ·ˆl = w ·ˆl = (B10) wherexη aregivenbyeq.A13. Usingtheunitvectorsˆl and 1 τ d 2 τ τd n τ 2π 2π ˆslτ⊥imwilaerclyanfowrrtihteeavnayriapbolient(cid:126)qin=spqa(cid:107)cˆleτa+s(cid:126)rq⊥=ˆlτ⊥x.(cid:107)Bˆlτy+inxtr⊥oˆldτ⊥ucainngd w1·ˆlτ⊥ = d(cid:16)1+ η1(cid:17) w2·ˆlτ⊥ =− d (B11) thequantity whered=(τ +1/η). Thereforewecandefine: A (q ,q )= lim 1 (cid:88)eıp(cid:126)n·q(cid:126) (B2) 2π (cid:18) h(cid:48)(cid:19) X (cid:107) ⊥ N→∞N n Q(h,h(cid:48))=khh(cid:48) ·ˆlτ = d h+ τ (B12) (cid:18) (cid:19) where we recall that p(cid:126) are points (cid:126)r of the two dimen- 2π η h sional periodic latticenwhich lie in an1snt2rip of width 2∆ = Q⊥(h,h(cid:48))=khh(cid:48) ·ˆlτ⊥ = d η+1 −h(cid:48) (B13) 7 Bymeansofeq.B4wecanthuswritetheintensitiesofthe where d = (τ + 1/η). By properly choosing the integers diffractedpointsas: h and h(cid:48), any real number can be arbitrarily well approxi- mated,showingthatthereciprocallatticeofaFLisdensein R,contrarilytoperiodiclatticeswhichexhibitadiscreterecip- I(q ,η)=|A(q )|2 =(cid:88)sin2(Q⊥(h,h(cid:48))∆)δ(q −Q(h,h(cid:48))) rocallattice. Moreoveritcanbeshown[3]thatthediffraction (cid:107) (cid:107) (Q (h,h(cid:48))∆)2 (cid:107) pattern has only pure point support, lacking of a continuous ⊥ h,h(cid:48) part (according to the classification of positive measures in (B14) theLebesgueclassification). where∆=τ(1+1/η)/2andweintroducetheexplicitdepen- denceoftheintensityontheparameterηwhichcharacterizes theFL.Asitcanbeseen,thediffractionspectrumconsistsof AppendixC:Brightestpeaks asetofsharppeakscenteredonadensesetofreciprocallat- ticepoints,asbychoosingtheappropriatevaluesofhandh’, 1. Conditionforbrightestpeaks anyqcanbeapproximatedwitharbitraryprecision. However, notallthesepeakshavethesameintensity. In order to find the set of points in reciprocal space cor- Infig.7weplotI(q )asgivenbyexpressionineq.B14and (cid:107) responding to a strong diffracted intensity for a FL, the fol- itsexpressioncalculatedexplicitlybyitsdefinitioneq.B1for lowing condition on the argument of the exponential in the a lattice of N = 300 points and η = 17/6. We can see that expressionforthediffractionpatternhastohold: asexpectedthepeaks’intensitiesarewellcapturedbyeq.B14 evenforfinitesystemsespeciallyforpeakscharacterizedbya significantintensity(¿0.2). Q (h,h(cid:48))≈0 (C1) ⊥ whichissatisfiedfor I q 1.0 h 1 ≈1+ . (C2) h(cid:48) η 0.8 Let us now consider the equivalence class [η ] and in par- 0 0.6 ticular the sequence xηn0. We can write η0 in the continued (cid:72) (cid:76) fractionrepresentation: 0.4 1 η =a + ≡[a ,a ,a ,a ,···] (C3) 0 0 a + 1 0 1 2 3 0.2 1 a2+a3+1... For a rational number the sequence of numbers a is finite, i q namely η0 = [a0,a1,··· ,an]. On the other hand, if η0 10 20 30 40 50 60 is irrational, it is possible to find a rational approximation within the wanted error by increasing the number of terms in its continued fraction representation. It is easy to see that Figure 7: (Color online) Comparison between the values of I(q ) (cid:107) η =1+1/η =[1,a ,a ,···]andingeneral usingthecutandprojectmethodandthedirectevaluationineq.B1 1 0 0 1 foralatticeofN =300pointsandη =17/6. Bluedotsarepoints η =[1,1,··· ,1,a ,a ,···]. (C4) k 0 1 correspondingtothevalueineq.B14whereasredcrossaregivenby (cid:124) (cid:123)(cid:122) (cid:125) eq.B1. k Withthisnotationisstraightforwardtoseethatregardlessof We now consider the (Fraunhofer) diffraction pattern of a thevalueofthegeneratorη0,thesequencesofanequivalence FLxη: class will tend to a Fibonacci sequence since limk→∞ηk = n [1,1,1,1,···] = τ [24]. Using the continued fraction nota- (cid:12) (cid:12)2 tion we can write a sequence of rational approximants to η0 I(q,η)=Nl→im∞N12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)eıxηnq(cid:12)(cid:12)(cid:12)(cid:12) . (B15) taosbae0,inat1eaag01e+rs1,thae2(aaba1o2aav01e++1c1)o+nad0i,ti·o·n·C. 2Siinscseatbisofithedhifanwdehc(cid:48)hhoaovsee n h=s +t andh(cid:48) =s wheres andt arethen-thapprox- n n n n n Thisquantityisimportantbecauseitgivesadirectexperimen- imants of η namely η ≈ s /t and can easily be derived 0 0 n n talaccesstothereciprocallatticeofourstructure.Weshallsee fromthecontinuedfractionrepresentationofη . 0 that this quantity is also encountered in the determination of It is worth stressing that if η is a rational number (η = 0 0 a(pseudo)energydispersionrelation[23]. InthecaseofFLs a/b,witha,b ∈ N)handh(cid:48) canbechosensuchthath/h(cid:48) = thevaluesofqatwhichanon-vanishingintensityisexpected (a+b)/a. Thus, at points q = Q(m(a+b),mb) = 2mπ m aregivenby: (m ∈ Z) we have that I(q ,η ) = 1. On the other hand, m 0 forirrationalη theconditionisneversatisfiedexactlybutwe 0 2π (cid:18) h(cid:48)(cid:19) can resort to the rational approximants of η to estimate the Q(h,h(cid:48))= h+ , (B16) 0 d τ positionsatwhichthebrightestpeaksappear. 8 2. Relationbetweenpositionsofbrightestpeaks ∆I q 0.25 Letxη0 andxη1 betwoFibonaccilatticesbelongingtothe n n sameequivalenceclassandtheirassociatedreciprocallattices 0.20 Q (h,h(cid:48))=2πd−1(h+h(cid:48)/τ),Q (h,h(cid:48))=2πd−1(k+k(cid:48)/τ) 0 0 1 1 respectively,wheredi =τ +1/ηi. Bydefininghn (kn)and 0.15(cid:72) (cid:76) h(cid:48) (k(cid:48)) as the numerator and denominator of the n − th n n rationalapproximantsof1+1/η0 (1+1/η1),thefollowing 0.10 relationsholdtruek = h +h(cid:48) andk(cid:48) = h . Byinserting n n n n n theserelationsintotheexpressionforQ1(k,k(cid:48))weget: 0.05 2π (cid:18) k(cid:48) (cid:19) 2π (cid:18) h (cid:19) q Q (k ,k(cid:48)) = k + n = h +h(cid:48) + n 10 20 30 40 50 60 70 1 n n d n τ d n n τ 1 1 2πτ (cid:18) h(cid:48) (cid:19) d τ 2π (cid:18) h(cid:48) (cid:19) = d1 hn+ τn = d01 d0 hn+ τn (Friegdurcero8s:s)(C(Io(lηo1rqo,nηl1in)e−)PIl(oqt,oηf0)(b)lfuoerdthootss)eτq−f1o(r1w−hicIh(qI,(ηq0,)η)0)an>d d τ 0.5showingthattheintensitiesofthebrightestpeaksofthediffrac- = d0 Q0(hn,h(cid:48)n)=η1Q0(hn,h(cid:48)n) tionpatternofaFLxηn1 generatedfromaFLxηn0 bymeansofthe 1 compositionrulearemoreintenseofthoseoftheoriginallatticeby whereinthelastlineweusedthefactthatd τ/d =η . This afactorofτ. 0 1 1 means that the lattice obtained by applying the composition ruleC(xηn0)=η1xηn1 hasbrightestpeaksatthesamepositions oftheoriginallatticeonlyrescaledbyafactoreta . 1 3. Relationbetweenintensitiesofbrightestpeaks From eq.B14 we can also estimate the relation between the intensities of the brightest peaks in the diffraction spec- trum of two FL belonging to the same class. Using the ex- samplewiththeaidofacomputer-controlledXYstageatcon- pression in eq.B14 and assuming k⊥∆ ≈ 0 we can write stantspeedof50µm/s. Thenominalprecisionofthetrans- sin2(k⊥∆)/(k⊥∆)2 −1 ≈ (k⊥∆)2/9. Using the condition lation stage is 0.5µm for the conditions used in this experi- fork⊥ ≈ 0andfollowingacalculationsimilartothattode- ment. A frequency doubled diode pumped Nd:YWO4 solid terminedrelationbetweenthepositionsofthebrightestpeaks state laser (Coherent Verdi V5) emitting a CW beam at 532 wefindthat(k⊥1∆1)2 =(k⊥0∆0)2/τ2. Thereforewehave nmhasbeenusedaslightsourceforDLW.Thebeamwassuit- ablyattenuatedbyaseriesofneutraldensityfiltersandsentto 1 I(η1q,η1)≈I(q,η0)+ τ(1−I(q,η0)), (C5) afocusingmicroscopeobjective(Olympus100X/0.80)sothat thepoweraftertheobjectivewassetat17mW.Thesubstrate meaning that the intensities of brightest peaks of the scaled usedtoengravetheopticalstructuresisaslabofphotorefrac- lattice are more intense of those of the original lattice by a tive lithium niobate doped with iron at the nominal concen- termproportionaltothedifferencebetweenthemaximumat- tration of 0.1 mol% in the melt. The sample was X-cut with tainableintensityandtheintensityoftheoriginallatticeinten- dimensions(X×Y×Z)1mm×8mm×13mmandthelines sities. werewrittenontheXface,byscanningalongtheYdirection Infig.8weplotthequantities(bluedots)τ−1(1−I(q,η0)) with an ordinarily polarized beam. This process can induce and (red cross) (I(η1q,η1) − I(q,η0)) for q such that extraordinaryrefractiveindexchangesaslargeas10−3inthe I(q,η0) > 0.5 and for lattices of N = 300 sites and η0 = written lines and therefore can be used to produce arbitrary 6/11andη1 =1+1/η0respectively. diffraction structures. The diffraction pattern of these struc- tures was measured with the help of a computer-controlled optical diffractometer in which the sample and the detector AppendixD:Experimentalsetup were mounted on two co-axial goniometers that were inde- pendently controlled by a computer [26]. An optical beam TotestexperimentallythediffractionfromFL’s,aseriesof producedbyaHe-Nelaserat632.8nmwithapowerof4mW quasi-periodic diffraction gratings have been prepared using wasexpanded,polarizedalongtheextraordinarydirectionand a photorefractive direct laser writing (DLW) technique [25]. finally transmitted through the sample surface, resulting in a Thistechniqueconsistsinscanningwithafocusedlaserbeam clearlyvisiblediffractionpattern. Thispatternwasmeasured aphotorefractivesample,engravingonitaseriesoflineswith by a Si photodiode and a lock-in amplifier and recorded on amodifiedrefractiveindexwithrespecttotherestofthesam- the computer as a function of the detector and of the sample ple. Thescanningmovementisperformedbytranslatingthe angle. 9 AppendixE:Experimentaldiffractionpatterns form for S(q) describes adequately the peak intensity in the wholerangeofmeasuredvalues(seefig. 9): 1. Structurefactor Inordertocomparetheexperimentaldatawiththetheoret- S(q)=S0e−λq−qq0 (E2) icalcalculationweneedtotakeintoaccountthatourgratings aremadeupofa(quasi)periodicrepetitionofaregionwitha where the parameters S ,λ,q are determined by a least 0 0 modifiedrefractiveindex,∆n(x). Thisleadstothefact,well square fit in the range q ∈ [0.5,3] µm−1 excluding the last knownfromstandarddiffractiontheory,thatthediffractedin- peaks because the corresponding momenta where compara- tensityinreciprocalspaceisproportionaltotheproductoftwo ble with a length scale of the order of the optical waveguide terms: afirstone,S(q)=(cid:12)(cid:12)(cid:82) ∆n(x)eıxqdx(cid:12)(cid:12)2whichdependson width. We also notice that all measured diffraction patterns thedetailedstructureoftherepeatedunitofthegrating(struc- drop almost to zero outside the interval q ∈ [−4,4] µm−1; turefactor)andasecondtermduetolatticegeometry,which this is due to the fact that our lines have a width determined isthetrueobjectofthisstudy: bythelaserwritingopticswhichisnotsmallerthan2µm,so thatourdiffractionpatterncannotprobe|q|>3µm−1. (cid:12) (cid:12)2 IR(q,η)=S(q)N12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)eıxηnq(cid:12)(cid:12)(cid:12)(cid:12) =S(q)I(q,η) (E1) 2. Comparisonwiththeoreticalpatterns n The structure factor modulates the intensity of the lattice In Fig. 10 we compare the theoretical diffraction pattern diffractionpattern,complicatingthecomparisonbetweenex- I(q,η) with the experimental data points for the grating ηb. periments and theory. In principle S(q) could be calculated 3 Theintensityoftheexperimentalpointshasbeenrescaledto byknowingthedetailsoftherefractiveindexprofilechanges take into account the contribution of the structure factor of produced by our technique. Here we used another approach the grating and the q axis of the experimental plot has been whichexploitsthefactoursamplesdifferonlyforthelinepo- rescaledinordertocompareitwiththesimulation,whichcon- sitionsequencex . Wecanusethereforetheperiodicgrating n siders FL’s with S = 1. A similar figure is obtained for the (Fig.9)tomeasurethefunctionS(q)directlyatthereciprocal caseηa =17/6. Theagreementisverysatisfactory: notonly latticepoints{qi }oftheperiodicgrating,whereI(qi )has 1 M M thepositionbutalsotheintensityofthediffractionpeaksare localmaxima. correctlyobtained,confirmingthatourapproachisreliable. I q 10(cid:45)4 I(q,η b) 10(cid:45)6 10(cid:45)8(cid:72) (cid:76) 10(cid:45)10 q q (cid:45)2 0 2 4 Figure 10: (Color online). Experimental diffraction pattern (solid Figure9: (Coloronline). Experimentaldiffractionpatternofthepe- redtopcurve)comparedwiththetheoreticaldiffractionpatternwith riodicgratingwithspacingL=23µm(blue,solidcurve)andfitof theinclusionofthestructurefactor(solidbluebottomcurve)fora thesatellitepeakintensitiesusingfunctionE2(red,dashedcurve). FibonaccigratingwithN = 300linesandL = 23 µmandS = 15µm,ηb =1.875. 3 We found that the following phenomenological functional [1] D.Shechtman,I.Blech,D.Gratias,andJ.W.Cahn,Phys.Rev. [4] C. Janot, Quasicrystals - A primer (second edition), Oxford Lett.53,1951(1984). UniversityPress(1994). [2] D.Levine,andP.J.Steinhardt,Phys.Rev.B34,596(1986). [5] Y. E. Kraus, and O. Zilberberg, Phys. Rev. Lett. 109, 116404 [3] M.Senechal,Quasicrystalsandgeometry,CambridgeUniver- (2012). sityPress(1995). [6] Y.E.Kraus,Y.Lahini,Z.Ringel,M.Verbin,andO.Zilberberg, 10 Phys.Rev.Lett.109,106402(2012). [18] Ch.Li,H.Cheng,R.Chen,Tian.Ma,Li-G.Wang,Yu.Song, [7] M.Verbin,O.Zilberberg,Y.E.Kraus,Y.Lahini,andY.Silber- andHai-Q.Lin,Appl.Phys.Lett.103,172106(2013). berg,Phys.Rev.Lett.110,076403(2013). [19] L.DalNegro(ed.)OpticsofAperiodicStructures: Fundamen- [8] M.Lohse,C.Schweizer,O.Zilberberg,M.Aidelsburger,andI. talsandDeviceApplications,PanStanfordUniversityPublish- Bloch,Nat.Phys.3584(2015). ing(2014). [9] K.Singh,K.Saha,S.A.Parameswaran,andD.M.Weld,Phys. [20] G.Radons,W.JustandP.Hussler(eds.)CollectiveDynamics Rev.A92,063426(2015). ofNonlinearandDisorderedSystems,Springer-Verlag,Berlin [10] R.W.Peng,Y.M.Liu,X.Q.Huang,F.Qiu,MuWang,A.Hu, -Heidelberg(2005). S.S.Jiang, D.Feng, L.Z.Ouyang, andJ.Zou, Phys.Rev.B [21] P.Buczek, L.Sadun, andJ.Wolny, ActaPhysicaPolonicaB, 69,165109(2004). 36,3(2005). [11] A.Lin,Y.K.Zhong,S.M.Fu,C.W.Tseng,andS.L.Yan,Opt. [22] J.-M.LuckPhys.Rev.B39,5834(1989). Expr.22(53)a880(2014). [23] J.-M. Gambaudo, and P. Vignolo, New J. 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Goodwin,arXiv:1508.05909.

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Equivalence classes of Fibonacci lattices and their similarity properties PDF

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Preview Equivalence classes of Fibonacci lattices and their similarity properties

EquivalenceclassesofFibonaccilatticesandtheirsimilarityproperties N. Lo Gullo,1,2 L. Vittadello,1 M. Bazzan,1,2 and L. Dell’Anna1,2 1Dipartimento di Fisica e Astronomia, Universita` degli Studi di Padova, Padova, Italy 2CNISM, sezione di Padova (Dated:July28,2016) Weinvestigate,theoreticallyandexperimentally,thepropertiesofFibonaccilatticeswitharbitraryspacings. Differently from periodic structures, the reciprocal lattice and the dynamical properties of Fibonacci lattices dependstronglyonthelenghtsoftheirlatticeparameters,evenifthesequenceoflongandshortsegment,the Fibonaccistring,isthesame. Inthisworkweshowthat,byexploitingaself-similaritypropertyofFibonacci stringsunderasuitablecompositionrule,itispossibletodefineequivalenceclassesofFibonaccilattices. We showthatthediffractionpatternsgeneratedbyFibonaccilatticesbelongingtothesameequivalenceclasscan 6 berescaledtoacommonpatternofstrongdiffractionpeaksthusgivingtothisclassificationaprecisemeaning. 1 Furthermoreweshowthat,throughthegaplabelingtheorem,gapsintheenergyspectraofFibonaccicrystals 0 belongingtothesameclasscanbelabeledbythesamemomenta(uptoaproperrescaling)andthatthelarger 2 gapscorrespondtothestrongpeaksofthediffractionspectra. Thisobservationmakesthedefinitionofequiv- l alence classes meaningful also for the spectral, and therefore dynamical and thermodynamical properties of u quasicrystals. Ourresultsapplytothemoregeneralclassofquasiperiodiclatticesforwhichsimilarityundera J suitabledeflationruleisinorder. 6 2 PACSnumbers: ] r e I. INTRODUCTION II. GENERALIZEDFIBONACCILATTICES h t o Inonedimension(1D)theparadigmofaquasicrystalisthe t. Sincethefirstexperimentalproofoftheexistenceofsolids Fibonaccilattice(FL).TheFLisa1Dlatticewhoseadjacent a pointshavedistancesbelongingtotheset{L,S},standingfor lacking of translational invariance, but exhibiting a discrete m LongandShortrespectively,whicharearrangedaccordingto Braggdiffractionspectrum[1],thestudyofquasicrystalsat- - tractedquitealotofattention.Theimpactofthisdiscoveryon agivensequence. Suchalatticecanbeconstructedbymeans d of the cut and project technique [3, 4, 21] thus obtaining for n thescientificcommunitywassuchthatin1992theformerdef- thecoordinatesofpointsontherealline[2](inunitsofS): o initionofcrystalhadtobemodifiedinordertoincludethose c structures whose diffraction pattern witnesses long range or- 1 (cid:106)n(cid:107) [ der yet lacking translational invariance [2–4]. More gener- xη =n−1+ (1) n η τ 2 ally, the study of quasiperiodic geometries has been recently v thesubjectofdifferentfieldsalldevotedtothepropagationof where n is a natural positive number,(cid:98)x(cid:99) is the integer part 9 wavesthroughquasiperiodicpotentials. Thespectralproper- of x and η = S/(L − S). The most common instance 9 tiesofquasicrystalshavebeenrecentlyusedtoengineertopo- (cid:0)√ (cid:1) found in literature is obtained for η = τ = 5+1 /2, 4 logicalpumpinginopticalwaveguides[5–7]andinultracold 7 the golden ratio. In this case the canonical FL (CFL) is ob- gases[8,9].Engeneeringofquasiperiodicstructureshavealso 0 tained, such that the lengths are (up to a simple rescaling): been employed in optical dielectric multilayers for resonant . L = 1 + 1/τ = τ, and S = 1. Nevertheless it is possi- 1 transmission [10], solar energy harvesting [11], plasmonics ble to construct Fibonacci lattices with η (cid:54)= τ (see App. A). 0 6 [12,13]andnonlinearoptics[14,15]. The distances ∆n = xηn+1 −xηn are either L = 1+1/η or 1 S =1andtheyarearrangedaccordingtotheFibonaccistring : Dynamical and transport phenomena in this kind of struc- (FS)LSLLSLSLLSLLSLSLLSLSL···.Thelatterisany v tures are also radically different compared to periodic me- wordmadeoftwoletters,LandS,obtainedbymeansofthe i X dia[16–20]. Forusualperiodicarrangements,dynamicaland substitutionruleS → LandL → LS startingfromtheletter r thermodynamicalpropertiesaredirectlyrelated,viatheBloch L. WenoticethataFSitselfisindependentontheparameter a theorem, to the geometry of the system. Quasiperiodic ge- ηanditonlydependsonthefactor1/τ. ometries,instead,lacksoftranslationalinvariancesothatadi- Conversely,givenaninfiniteFS,acompositionrule(LS → rectrelationbetweentheirstructureandtheirdynamicalprop- L(cid:48) and L → S(cid:48)) can be defined such that the old and the erties is not generally known. It would be therefore very in- newstringsarethesameduetothepeculiarpropertiesofthe terestingtofindasortofclassificationenablingonetogroup Fibonacci strings, as shown in Fig. 1. For the special case togetherdifferentaperiodicsystemsonthebasisofsomesim- η = τk, with k a non-vanishing integer i.e. for the canoni- ilaritybetweentheirgeometricarrangements.Inthispaperwe cal FL, this leads to a peculiar property: the new FL can be attempttodefinesuchaclassification,showingthatquasiperi- rescaledtotheoriginalone. Thiscaseisthemostcommonly odicstructureswhosegeometryisrelatedbyasuitablemath- encountered in literature, accompanied by the statement that ematicaltransformationsharethemaincharacteristicsoftheir the CFL is self-similar. It should be stressed however that reciprocallatticeandoftheirpseudo-bandstructure. this is not true for the general case η (cid:54)= τk. In this case, a 2 for irrational η we can resort to its continued fraction repre- L S L L S L S L L S L sentationinordertosetthewantedprecisiontotheabovecon- dition. L' S' L' L' S' L' S' LetusnowconsidertwoFLsbelongingtothesameequiv- alenceclassxηn0 andxηn1,withη1 =1+1/η0. Bydefininghn Figure1: Thecompositionrules(LS → L(cid:48), L → S(cid:48))onasemi- (kn)andh(cid:48)n(kn(cid:48))asthenumeratoranddenominatorofthen- infinite Fibonacci lattice reproduces another Fibonacci lattice with thrationalapproximantsof1+1/η (1+1/η ),thefollowing 0 1 differentlatticeparameters. relationshold: k = h +h(cid:48) andk(cid:48) = h . Thepositionof n n n n n thebrightestpeaksoftheFLxη1 arethenrelatedtothoseof n theFLxη0 by: n non-canonical Fibonacci lattice and the one obtained by ap- plyingthecompositionrulearecharacterizedbytwodifferent Q1(kn,kn(cid:48))=η1Q0(hn,h(cid:48)n) (4) lenghtratiosη andη becauseL(cid:48)/L (cid:54)= S(cid:48)/S. Thereforethe 1 2 In other word, althought the two Fibonacci lattices xη0 and newlatticecannotbetransformedintotheoldonebyasimple n rescaling. C(xηn0) = η1xηn1 cannot be rescaled one over the other (for the general case η (cid:54)= τ), their brightest peak pattern can, as We call C the operator corresponding to the effect of the aconsequenceofthefactthattheyarerelatedbythecompo- compositionruleontheFLxη. Itisnotdifficulttoshowthat n sition rule. Also the intensities of the brightest peaks can be C(xηn) = η1xηn1, whereη1 = 1+1/η: thecompositionrule relatedasI(η q,η )≈I(q,η )+ 1(1−I(q,η ))(forqsuch maps a FL xη into another FL, characterized by a new ratio 1 1 0 τ 0 n thatI(q,η ) > 0.5)showingthatthepeaksofthescaledlat- η and rescaled by η (see Fig. 1). If η = τ then η = τ 0 1 1 1 tice are even brighter than those of the original lattice. This (recall that τ2 −τ −1 = 0) and therefore the CFL is self- drives to the important conclusion that FL belonging to the similar. Ifthecompositionruleisappliedk times, theinitial sameequivalenceclasshavediffractionspectracharacterized FLismappedinto bythesamepatternofbrightestpeaks,andare,inthissense, (cid:32) k (cid:33) (cid:18) (cid:19) similar. C(k)(xηn)= (cid:89)ηi xηnk = Fk+1+ Fηk xηnk (2) i=1 III. SIMILARITYOFDIFFRACTIONPATTERNS where η = 1+1/η (η = η) and F are the Fibonacci i i−1 0 k numbers. Withthehelpofthisconceptwedefineequivalence Toquantifythedegreeofsimilaritybetweenthetwospec- classes for FLs by means of the following equivalence rela- tra, we use the Kullback-Leibler divergence (KLD), a quan- tion: tity useful to compare two distributions (normalized to unity over a common support). Let us consider the diffraction Definition - Two Fibonacci lattices xηna and xηnb are equiva- spectra I(q,ηα) and I(q,ηβ) of two arbitrary FL’s charac- lent(xηna ∼ xηnb)iftheyarelinked,uptoaproperrescaling, terized by ηα (cid:54)= ηβ. We define the normalized spectrum: bymeansofthecompositionruleC. Thelatticewiththemini- P(νq,η) = I(νq,η)/(cid:82)∞dkI(νk,η) where we introduced 0 mumη0suchthat1/(η0−1)isfiniteandpositiveiscalledthe ascalingparameterν. generatoroftheequivalenceclasswhichisdenotedby[η0]. TheKLDisdefinedas: Asimplewayoflabelingtheelementsofagivenequivalence D(η ,η ,ν)=(cid:90) ∞dkP(k,η )log(cid:18) P(k,ηα) (cid:19). (5) classisbymeansofthecontinuedfractionrepresentationfor α β α P (νk,η ) 0 β the{η }. i Bydefinitiononehasthatthemoresimilarthetwodiffraction Because of this, for quasiperiodic structures it is of lim- spectra, the smaller the value of the KLD. We will use it to ited practical utility to talk about the support of the diffrac- measureif,forgivenη andη ,thereexistascalingparameter tion pattern. It is more meaningful to describe the diffrac- α β ν forwhichthetwospectralooksimilar. tionspectrum(andthereciprocallattice)intermsofthepeaks In Fig. 2 a) and b) we plot 1/D(ηα,η,ν) comparing two whicharesignificantlyclosetoone,whichwillbereferredto 0 generators corresponding to ηa = 6/11 and ηb = 1/6 with as brightest peaks. By means of the cut and project method 0 0 FLs obtained from them by applying the composition rule outlined in App. A it is possible to show (see App. B) that the intensity I(q,η) at points q = Q(h,h(cid:48)) is given C(nα) respectively na = 1 and nb = 3 times. The inten- by sinc2(Q (h,h(cid:48))∆), where Q (h,h(cid:48)) = 2πd−1(h(1 + sities I(q,η) are evaluated by means of eq.B15 for lattices ⊥ ⊥ with N = 300 points. The maxima (minima of D) in the 1/η)−1−h(cid:48))and∆=τ(η/(η+1))/2. Thereforethebright- two figures correspond to (η,ν) = (ηa,ηa) and (η,ν) = estpeaksarefoundforpairs(h,h(cid:48))suchthatQ (h,h(cid:48)) ≈ 0 1 1 ⊥ (ηb,ηbηbηb) respectively, in agreement with eq. (4). This andthusfor 1 1 2 3 shows that two FL’s produce a similar diffraction pattern if h 1 andonlyiftheycanberelatedviaEq. (2)andthereforeonly =1+ . (3) h(cid:48) η iftheybelongtothesameequivalenceclass. In order to test our results on a real case, we performed a Sincehandh(cid:48) areintegers,theaboveconditioncanbesatis- diffraction experiment on two quasiperiodic diffraction grat- fiedexactlyonlyifη isarationalnumber. Ontheotherhand ings prepared using a photorefractive direct laser writing 3 D Figure 3: (Color online). a) Comparison between the theoretical diffractionpatternfortheFLxη0a (ηa = 0.5454, solidbluecurve) andtheexperimentaloneproducnedby0aFLC(cid:16)xη0a(cid:17)(solidredcurve, n N = 300 lines and L = 23 µm and S = 17 µm, ηa = 2.83). 1 ThetheoreticalspectrumhasbeenrescaledinaccordancewithEq. (4) and corrected for the structure factor contribution. b) The in- verse of KL divergence (D−1(ηa,η,ν)) between the (normalized Figure 2: (Color online) Inverse of KL divergence 1/D(ηa,η,ν) 1 0 over the interval) experimental diffraction pattern and the theoreti- comparing the diffraction spectra of the generator of a given class caldiffractionpatternsfordifferentgeneratorsanddifferentscaling. xηn0 with another Fibonacci lattice, for two choices of η0. a)η0a = Here∆ηa =η−ηaand∆ν =ν−1/ηa. Themaximum(minima 6/11;b)ηb =1/6.Here∆ηa =η−ηa,∆νa =ν−ηaand∆ηb = 0 0 0 1 1 ofD)isat(η,ν)=(ηa,1/ηa). η−ηb,∆νb =ν−ηbηbηb.Themaxima(minimaofD(η ,η ,ν)) 0 0 1 1 2 3 α β areobtainedat∆ηa,b = 0and∆νa,b = 0,indicatingthatthetwo spectra with the higest degree of similarity corresponds to lattices ηaxη1a =C(cid:16)xη0a(cid:17)andηbηbηbxη3b =C(3)(xη0b)respectively,i.e.the arangeofη0 andscalingfactorν. InFig. 3b)weshowitex- 1 n n 1 2 3 n n plicitlyforthegratingwithηaanditisclearthatthemaximum 1 firstandthethirdelementoftherespectiveequivalenceclasses. c) ofD−1 (minimumofD)isfoundatη = ηα andν = η−1. DirectcomparisonofthetwodiffractionspectrafortwoFLsxηn0aand Similarresultsareobtainedforthegrat0ingηb0. 1 ηaxη1a.ThemostprominentpeaksofthediffractionpatternI(q,ηa) 1 n 0 correspondwiththoseofthe(rescaled)spectrumI(ηaq,ηa). 1 1 IV. ENERGYSPECTRACOMPARISON We have therefore shown that all the FL belonging to the (DLW) technique [25, 26]. We used three gratings made up sameequivalenceclasshavediffractionspectrathat,although ofN =300linesallwritteninthesamesubstrate: (a)aperi- not equals, are characterized by a similar pattern of bright odicgratingwithspacingL=23µm;twoFibonaccigratings with (b) L = 23µm and S = 17µm (ηa = 17/6) and (c) peaks. Thisfindingisofcrucialimportancenotonlyinscat- L = 23µmandS = 15µm(ηb = 15/8)1respectively. Sofar tering phenomena but also in transport ones. In fact in a re- 3 cent work [23] a method to unambiguously link the gaps in weconsideredpointlattices,butrealstructuresareconstitued the integrated density of states to the brightest peaks in the bysomephysicalentity(basis)arrangedonthepointsofour diffraction pattern of the underlying potential has been pro- quasi-periodic Fibonacci lattice. (For a detailed description posed.Thisismoregeneralhasithasbeenshowninaseminal of the experimental set up see App. E). For these cases, the paperbyLuck[22]. LetusconsiderforexampletheHamilto- diffraction pattern is given by the sum in Eq. (B15) multi- nianforaparticleina1Dlattice: plied by the square modulus of a structure factor. The latter, in general, does not posses any scaling property and there- (cid:126)2 d2 foreitisnecessarytocorrectforitwhencomparingdifferent Hˆ = − +V(x) (6) 2 dx2 lattices. We did this experimentally by using the data of the (cid:90) periodicgratingtoextractaphenomenologicalexpressionfor V(x) = −V dyf(x−y)(cid:88)δ(y−x ) (7) 0 n the structure factor as a function of q. In Fig. 3 a) we com- n pare the experimental data relative to the grating ηb with the 1 theoretical diffraction pattern obtained from the generator of where x are the local minima of the potential and f(x) is n the corresponding equivalence class, ηb = 1/6. We observe introduced to account for the detailed shape of the potential 0 that,oncethespectrahavebeenrescaledinqfollowingeq.(4) minima (V > 0). We will consider the case x = xη ac- 0 n n andcorrectedinordertotakeintoaccountthestructurefactor cordingtothequasi-periodicsequenceofEq.1. InRef.[23] contributiontotheintensityofthepeaks,themostprominent it has been shown that it is possible to label the energy gaps diffraction features of the generator can be found in the ex- bymeansofthebrightestpeaksofthediffractionspectrum.In perimentaldataatthecorrectq positions. Thedegreeofsim- particularonehastoconsiderthepseudo-momentaqatwhich ilaritybetweenthespectrumofthegeneratorandtheexperi- the square of the Fourier transform of V(x) acquires values mentaloneisconfirmedbytheKLdivergenceDbetweenthe greater than a given threshold. This effectively corresponds experimentaldatapointsandthetheoreticaldiffractionspec- tochoosewhichfreestatesareeffectivelycoupledbythepo- trum(withtheinclusionofthestructurefactor)calculatedfor tential and, therefore, where wider gaps open in the single 4 shownthatitispossibletogroupdifferentFibonaccilattices intoequivalenceclasseswhoseelementssharethemainstruc- tural and dynamical properties as witnessed by their diffrac- tionspectraandtheenergygaps. Theseresultsshowthatthe concept of equivalence classes for FLs has not only a geo- metrical meaning but also an important role in the scatter- I ing,dynamicalandthermodynamicalpropertiesofthesystem, contained in the energy spectrum. It is worth stressing once again that this is a consequence of the self-similarity of FSs 0. under the composition rule and that FLs belonging to differ- entequivalenceclassescannotberescaledoneovertheother. Thegeneratorofaclassis,inthissense,thesimpleststructure giving a diffraction pattern which contains the main features common to all of the other elements of the class. Although Figure 4: (Color online). Plot of the energy level spacings of the wefocusedontheFibonaccilattices,ourargumentsapplyto Hamiltonian in Eq.(7) for a potential V(x) having minima at the themoregeneralclassofquasicrystalsforwhichdeflationor points of FLs xη with (blue dots) η = ηa, (red crosses) ηa. Be- lowwecomparenthegapswiththebrightest0peaksofI(q/ηa,1ηa). inflationrulescanmaptheinitiallatticesintoasimilarones. 1 1 particlespectrum. OneoftheresultspresentedinRef.[23]is Acknowledgments thatthisisequivalenttosetathresholdtotheintensityofthe peaks in the Bragg spectrum of the lattice. This can be eas- NL and LD acknowledge financial support from MIUR, ilyseenbyconsideringtheFouriertransformofthepotential throughFIRBProjectNo. RBFR12NLNA 002. LVandMB V(x),namelyV(q) = (cid:82) eıxqV(x)dxwhosesquaremodulus acknowledge financial support from Universit degli studi di isgivenby: Padova through Chip & CIOP project No. CPDA120359. The authors are thankful to Prof. Camilla Ferrante and Dr. |V(q)|2 =S(q)I(q,η) (8) Nicola Rossetto, from the Dipartimento di Chimica, Univer- whereS(q)isthesquareoftheFouriertransformoff(x)and sit di Padova, for providing access to the direct laser writing I(q,η)isgivenbyEq.B15.Itisclearfromwhatshownabove setup. The authors thank J. Settino for providing the energy andconfermedbytheexperimentonthediffractionpatterns, spectraofaparticleinaFibonaccilikelattice. that,apartfromthecontributionoftheactualformofthepo- tential(whichplaysaroleanalogoustothestructurefactorin diffractionexperiments),theenergypseudo-bandstructurein AppendixA:GeneralizedFibonaccilatticesfromcutand reciprocalspacehasthesameshape(uptoarescaling)forthe projectmethod lattices beloging to a given class. As an example, we com- puted the spectra of Eq. (7) in the case of Gaussian wells, The Fibonacci lattices we considered in the main text can namely f(x) = e−x2/2σ2 for a system with N = 100 min- beconstructedbymeansofthecutandprojecttechnique.One ima and x = xη with η = ηa,ηa and η = ηb,ηb. We possibleconstructionhasbeenpresentedinref.[21]Weprefer n n 0 1 0 3 choose V = 12, σ = 0.1. In Fig. 4 we plot the energy toresorttoamorestandardoneandinwhatfollowswewill 0 level spacing for lattices characterized by ηa = 6/11 (blue generalizetheonegiveninref. [3]. dots) and ηa = 17/6 (red crosses) both b0elonging to the Letusintroduceatwo-dimensionalperiodiclatticeIp and 0 2 equivalence class [η0a]. The momenta on the x-axis serve as its lattice vectors e1 and e2 such that any point of the lat- areferencewithrespecttothefreeparticledispersionrelation tice can be written as p = n1e1 + n2e2 with n1,n2 ∈ Z. ((cid:15)k = k2/2,V(x) = 0) to show where the potential V(x) Furthermore we introduce the line lτ whose unit vector is opens the gaps. After rescaling the momenta for the lattices ˆlτ = (cos(θτ),sin(θτ)) and the unit vector orthogonal to it withη = ηa byν = ηa wecanclearlyseethatthegapsap- ˆl⊥ = (sin(θ ),−cos(θ )) such that tan(θ ) = τ−1 where 1 a 1 τ √τ τ τ pearatthesamepoints. Ontheotherhandthesepointscorre- τ = (1 + 5)/2. The canonical Fibonacci lattice is con- spondtothebrightestpeaks,whereI(q/ηa,ηa)calculatedby structed by projecting on the line l the points of a square 1 1 τ Eq.(B15)issizeable.Similarresultsareobtained(notshown) lattice(e ·e = 0,|e | = |e |)thepointswhoseVonro¨ıcell 1 2 1 2 for the equivalence class [ηb] with ηb = 1/6 by considering iscutbythelineitself. Letusnoticethatwiththisprocedure 0 0 thetwolatticescharacterizedbyηb andηb =15/8,underthe thedifferentpointsareunambiguouslynumberedontheline 0 3 scalingνb =η1bη2bη3b. lτ onceanoriginandadirectionhavebeenchosen.Wearego- ingtoconstructourFibonaccilatticesusingthisdefinitionbut allowingthetwo-dimensionallatticetobegenericasinFig.5. V. CONCLUSIONS Nevertheless we will see that in order to obtain a Fibonacci lattice, namely a one-dimensional set of points whose dis- In conclusion, we investigated the diffraction spectra of tances are distributed according to the Fibonacci strings and FL’s in the general case η = S/(L−S) (cid:54)= τ. We have with the wanted ratio between long and short segments, we 5 willhavetorestrictthesetoftheallowedtwodimensionallat- tices. Followingthediscussioninref.[3]inorderfortheline γˆl tocuttheVonro¨ıcellcenteredatpointpithastointersect τ the secondary diagonal of the cell (joining the northwest to southeastpointofthecell). Thediagonalslieonlinesparallel toδ(e −e )andwhosepointsaregivenbyδ(e −e )+ne 1 2 1 2 1 with m ∈ Z. Their intersection with the line γˆl occurs at √ τ points(cid:0)n, n(cid:1)awherea=(τ2/ 1+τ2)A /(e −e )·ˆl⊥ τ τ2 u.c. 1 2 τ andA =(e ∧e )·zˆisthe(oriented)areaoftheunitcell u.c. 1 2 ofthelattice. Figure 5: (Color online) Construction of a generalized Fibonacci lattice from a 2D periodic lattice by means of the cut and project method.Bluedotsaretheprojectionofpointsofthe2Dlatticewhose ThisintersectionpointsareinsidetheVonro¨ıcellcentered Vonro¨icellsarecutbythelinelτ. atpoint(u,v)ifandonlyif |(e −e )·xˆ| |(e −e )·xˆ| u− 1 2 < naτ−1 <u+ 1 2 (A1) 2 2 |(e −e )·yˆ| |(e −e )·yˆ| v− 1 2 < naτ−2 <v+ 1 2 (A2) 2 2 On the other hand each point of the lattice can be written as projected.Thuswecanwriteu=n (e −e )·xˆ+ne ·xˆand 1 1 2 2 n e +n e andn +n =nbecauseitisthen-thpointtobe v =n (e −e )·yˆ+ne ·yˆandaboveinequalitiesbecome 1 1 2 2 1 2 1 1 2 2 (cid:16) s (cid:17) (cid:16) s (cid:17) n − x (e −e )·xˆ< n(aτ−1−e ·xˆ) < n + x (e −e )·xˆ (A3) 1 2 1 2 2 1 2 1 2 (cid:16) s (cid:17) (cid:16) s (cid:17) n − y (e −e )·yˆ< n(aτ−2−e ·yˆ) < n + y (e −e )·yˆ (A4) 1 2 1 2 2 1 2 1 2 wheres =Sign((e −e )·xˆ)andsimilarlyfors .Bymeans coordinatesonthethelinel : x 1 2 y τ of the expression for a it is easy to prove that (a τ−1 −e · 2 xˆ)/(e −e )·xˆ = (a τ−2−e ·yˆ)/(e −e )·yˆandthus 1 2 2 1 2 (cid:22) (cid:23) thetwoinequalitiesareequivalenttotheinequality: n x(cid:48) =ne ·ˆl +(e −e )·ˆl (A7) n 2 τ 1 2 τ β (cid:18) (cid:19) (cid:18) (cid:19) 1 n 1 n1− 2 < β < n1+ 2 (A5) Bynormalizingwithrespecttoe2·ˆlτ weeventuallyobtainthe one-dimensionallatticeofpoints e ·ˆl⊥ 1 β =1− 1 τ =1+r (A6) e ·ˆl⊥ τsin(α)−cos(α) 2 τ (cid:22) (cid:23) 1 n x =n+ (A8) where cos(α) = e1 ·e2/(|e1||e2|) and r = |e1|/|e2|. Be- n η β iinngeqnua1liatniesinttoegbeersnautimsfibeedritshethoatnlny1p=oss(cid:98)inβbi(cid:99)liwtyhfeorer t(cid:98)hxe(cid:99)aibsothvee η−1 =(cid:32)e1·ˆlτ −1(cid:33). (A9) integerpartofx. Afterprojectingontol ,then-thpointhas e ·ˆl τ 2 τ 6 In order for the above to be a Fibonacci lattice we require |(e − e ) · ˆl⊥| around the line γˆl . It is easy to see that 1 2 τ τ β = τ which is the case for τr = (τsin(α)−cos(α)) and A(q ) = A (q ,0). We therefore turn to the calculation of (cid:107) X (cid:107) thusη =(τ +tan(α))/((τ −1)tan(α)−τ2). Moreoverwe thelatter. Byintroducingthemassdensityofthetwodimen- (cid:80) have to require that r > 0 and η > 0 which is the case for sional lattice ρ((cid:126)r) = δ((cid:126)r −(cid:126)r ) and its Fourier tan−1(2τ +1) < α < tan−1(−τ)+π. As it can be seen transformρ((cid:126)r)=(cid:82) dk dmk1me−2ı(cid:126)r·(cid:126)kρ˜((cid:126)k)mw1emc2anwrite: ⊥ (cid:107) fromfigs.6foranygivenη >0therecorrespondapair(r,α): 1 1 (cid:90) A (q ,q ) = lim dk dk (B3) ητ +1 X (cid:107) ⊥ L→∞L2∆ ⊥ (cid:107) tan(α) = τ2 (A10) (cid:90) ∆ (cid:90) ∞ η−τ dx dx eı(cid:126)r·(q(cid:126)−(cid:126)k) ρ˜((cid:126)k) ⊥ (cid:107) (τtan(α)−1) −∆ −∞ r = (A11) (cid:112) τ 1+tan2(α) Theaboveintegralscanbecalculated: andthereforeatwodimensionallatticewhoseprojectionon A(q )=A (q ,0)=(cid:90) dk sin(k⊥∆)(cid:90) dk ρ˜((cid:126)k)δ(k −q ) thelinelτ returnsthewantedFL: (cid:107) X (cid:107) ⊥ k⊥∆ (cid:107) (cid:107) (cid:107) (B4) 1 (cid:106)n(cid:107) xη =n−1+ (A12) which expresses the fact that the diffraction pattern of an in- n η τ finite projected quasicrystal is the convolution of the Dirac (cid:32) (cid:33) e ·ˆl combformedbytheperiodichigherdimensionalperiodiclat- η−1 = 1 τ −1 , (A13) ticewiththesincfunctionintheorthogonalspace. e ·ˆl 2 τ Inparticularρ˜((cid:126)k)isaDiraccombpeakedatpointsk = hh(cid:48) whereweshiftedthewholelatticeinorderforthefirstpoint hw1+h(cid:48)w2whereweintroducedthereciprocallatticevectors tohavecoordinatex =0onthelinel . forthedualofthetwo-dimensionalperiodiclatticeIp: 1 τ 2 2π w = (e −e ·eˆ eˆ ) (B5) 1 l 1 1 2 2 1 2π w = (e −e ·eˆ eˆ ) (B6) 2 l 2 2 1 1 2 (B7) whereeˆ =e /|e |andl =|e |2−|e ·eˆ |2andsimilarlyfor i i i 1 1 1 2 l . Itiseasytocheckthatw ·e = 2πδ . Inwhatfollows 2 i j ij Figure6: (Coloronline)a)Valuesofr =|e1|/|e2|asafunctionof we assume that units are scaled such that e2 · ˆlτ = 1. In η. The red dot corresponds to the point (η,r) = (τ,1) for which ordertoevaluatetheparallelandperpendicularcomponentsof thecanonicalFibonaccilatticeisobtained. b)Valuesoftheangleα vectorsbelongingtothereciprocalspaceweneedtoevaluate betweene1ande2asafunctionofη.Thereddotcorrespondstothe wi·ˆlτ andwi·ˆlτ⊥. Inordertodosoitisusefultorewritethe point(η,α) =(τ,π/2)forwhichthecanonicalFibonaccilatticeis vectorse aslinearcombinationsofˆl andˆl⊥bymeansofthe obtained. i τ τ expressionsforη,βandtherelationbetweentanαandη.We thusobtain: (cid:18) (cid:19) (cid:18) (cid:19) AppendixB:Diffractionpattern e = 1+ 1 ˆl + 1 1+ 1 ˆl⊥ (B8) 1 η τ τ η τ Weareinterestedinthecalculationofthequantity: (cid:18) 1(cid:19) e = ˆl − 1+ ˆl⊥ (B9) 2 τ η τ A(q(cid:107))=Nl→im∞N1 (cid:88)eıxηnq(cid:107) (B1) Itisnoweasytocheckthat: n 2π 2π w ·ˆl = w ·ˆl = (B10) wherexη aregivenbyeq.A13. Usingtheunitvectorsˆl and 1 τ d 2 τ τd n τ 2π 2π ˆslτ⊥imwilaerclyanfowrrtihteeavnayriapbolient(cid:126)qin=spqa(cid:107)cˆleτa+s(cid:126)rq⊥=ˆlτ⊥x.(cid:107)Bˆlτy+inxtr⊥oˆldτ⊥ucainngd w1·ˆlτ⊥ = d(cid:16)1+ η1(cid:17) w2·ˆlτ⊥ =− d (B11) thequantity whered=(τ +1/η). Thereforewecandefine: A (q ,q )= lim 1 (cid:88)eıp(cid:126)n·q(cid:126) (B2) 2π (cid:18) h(cid:48)(cid:19) X (cid:107) ⊥ N→∞N n Q(h,h(cid:48))=khh(cid:48) ·ˆlτ = d h+ τ (B12) (cid:18) (cid:19) where we recall that p(cid:126) are points (cid:126)r of the two dimen- 2π η h sional periodic latticenwhich lie in an1snt2rip of width 2∆ = Q⊥(h,h(cid:48))=khh(cid:48) ·ˆlτ⊥ = d η+1 −h(cid:48) (B13) 7 Bymeansofeq.B4wecanthuswritetheintensitiesofthe where d = (τ + 1/η). By properly choosing the integers diffractedpointsas: h and h(cid:48), any real number can be arbitrarily well approxi- mated,showingthatthereciprocallatticeofaFLisdensein R,contrarilytoperiodiclatticeswhichexhibitadiscreterecip- I(q ,η)=|A(q )|2 =(cid:88)sin2(Q⊥(h,h(cid:48))∆)δ(q −Q(h,h(cid:48))) rocallattice. Moreoveritcanbeshown[3]thatthediffraction (cid:107) (cid:107) (Q (h,h(cid:48))∆)2 (cid:107) pattern has only pure point support, lacking of a continuous ⊥ h,h(cid:48) part (according to the classification of positive measures in (B14) theLebesgueclassification). where∆=τ(1+1/η)/2andweintroducetheexplicitdepen- denceoftheintensityontheparameterηwhichcharacterizes theFL.Asitcanbeseen,thediffractionspectrumconsistsof AppendixC:Brightestpeaks asetofsharppeakscenteredonadensesetofreciprocallat- ticepoints,asbychoosingtheappropriatevaluesofhandh’, 1. Conditionforbrightestpeaks anyqcanbeapproximatedwitharbitraryprecision. However, notallthesepeakshavethesameintensity. In order to find the set of points in reciprocal space cor- Infig.7weplotI(q )asgivenbyexpressionineq.B14and (cid:107) responding to a strong diffracted intensity for a FL, the fol- itsexpressioncalculatedexplicitlybyitsdefinitioneq.B1for lowing condition on the argument of the exponential in the a lattice of N = 300 points and η = 17/6. We can see that expressionforthediffractionpatternhastohold: asexpectedthepeaks’intensitiesarewellcapturedbyeq.B14 evenforfinitesystemsespeciallyforpeakscharacterizedbya significantintensity(¿0.2). Q (h,h(cid:48))≈0 (C1) ⊥ whichissatisfiedfor I q 1.0 h 1 ≈1+ . (C2) h(cid:48) η 0.8 Let us now consider the equivalence class [η ] and in par- 0 0.6 ticular the sequence xηn0. We can write η0 in the continued (cid:72) (cid:76) fractionrepresentation: 0.4 1 η =a + ≡[a ,a ,a ,a ,···] (C3) 0 0 a + 1 0 1 2 3 0.2 1 a2+a3+1... For a rational number the sequence of numbers a is finite, i q namely η0 = [a0,a1,··· ,an]. On the other hand, if η0 10 20 30 40 50 60 is irrational, it is possible to find a rational approximation within the wanted error by increasing the number of terms in its continued fraction representation. It is easy to see that Figure 7: (Color online) Comparison between the values of I(q ) (cid:107) η =1+1/η =[1,a ,a ,···]andingeneral usingthecutandprojectmethodandthedirectevaluationineq.B1 1 0 0 1 foralatticeofN =300pointsandη =17/6. Bluedotsarepoints η =[1,1,··· ,1,a ,a ,···]. (C4) k 0 1 correspondingtothevalueineq.B14whereasredcrossaregivenby (cid:124) (cid:123)(cid:122) (cid:125) eq.B1. k Withthisnotationisstraightforwardtoseethatregardlessof We now consider the (Fraunhofer) diffraction pattern of a thevalueofthegeneratorη0,thesequencesofanequivalence FLxη: class will tend to a Fibonacci sequence since limk→∞ηk = n [1,1,1,1,···] = τ [24]. Using the continued fraction nota- (cid:12) (cid:12)2 tion we can write a sequence of rational approximants to η0 I(q,η)=Nl→im∞N12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)eıxηnq(cid:12)(cid:12)(cid:12)(cid:12) . (B15) taosbae0,inat1eaag01e+rs1,thae2(aaba1o2aav01e++1c1)o+nad0i,ti·o·n·C. 2Siinscseatbisofithedhifanwdehc(cid:48)hhoaovsee n h=s +t andh(cid:48) =s wheres andt arethen-thapprox- n n n n n Thisquantityisimportantbecauseitgivesadirectexperimen- imants of η namely η ≈ s /t and can easily be derived 0 0 n n talaccesstothereciprocallatticeofourstructure.Weshallsee fromthecontinuedfractionrepresentationofη . 0 that this quantity is also encountered in the determination of It is worth stressing that if η is a rational number (η = 0 0 a(pseudo)energydispersionrelation[23]. InthecaseofFLs a/b,witha,b ∈ N)handh(cid:48) canbechosensuchthath/h(cid:48) = thevaluesofqatwhichanon-vanishingintensityisexpected (a+b)/a. Thus, at points q = Q(m(a+b),mb) = 2mπ m aregivenby: (m ∈ Z) we have that I(q ,η ) = 1. On the other hand, m 0 forirrationalη theconditionisneversatisfiedexactlybutwe 0 2π (cid:18) h(cid:48)(cid:19) can resort to the rational approximants of η to estimate the Q(h,h(cid:48))= h+ , (B16) 0 d τ positionsatwhichthebrightestpeaksappear. 8 2. Relationbetweenpositionsofbrightestpeaks ∆I q 0.25 Letxη0 andxη1 betwoFibonaccilatticesbelongingtothe n n sameequivalenceclassandtheirassociatedreciprocallattices 0.20 Q (h,h(cid:48))=2πd−1(h+h(cid:48)/τ),Q (h,h(cid:48))=2πd−1(k+k(cid:48)/τ) 0 0 1 1 respectively,wheredi =τ +1/ηi. Bydefininghn (kn)and 0.15(cid:72) (cid:76) h(cid:48) (k(cid:48)) as the numerator and denominator of the n − th n n rationalapproximantsof1+1/η0 (1+1/η1),thefollowing 0.10 relationsholdtruek = h +h(cid:48) andk(cid:48) = h . Byinserting n n n n n theserelationsintotheexpressionforQ1(k,k(cid:48))weget: 0.05 2π (cid:18) k(cid:48) (cid:19) 2π (cid:18) h (cid:19) q Q (k ,k(cid:48)) = k + n = h +h(cid:48) + n 10 20 30 40 50 60 70 1 n n d n τ d n n τ 1 1 2πτ (cid:18) h(cid:48) (cid:19) d τ 2π (cid:18) h(cid:48) (cid:19) = d1 hn+ τn = d01 d0 hn+ τn (Friegdurcero8s:s)(C(Io(lηo1rqo,nηl1in)e−)PIl(oqt,oηf0)(b)lfuoerdthootss)eτq−f1o(r1w−hicIh(qI,(ηq0,)η)0)an>d d τ 0.5showingthattheintensitiesofthebrightestpeaksofthediffrac- = d0 Q0(hn,h(cid:48)n)=η1Q0(hn,h(cid:48)n) tionpatternofaFLxηn1 generatedfromaFLxηn0 bymeansofthe 1 compositionrulearemoreintenseofthoseoftheoriginallatticeby whereinthelastlineweusedthefactthatd τ/d =η . This afactorofτ. 0 1 1 means that the lattice obtained by applying the composition ruleC(xηn0)=η1xηn1 hasbrightestpeaksatthesamepositions oftheoriginallatticeonlyrescaledbyafactoreta . 1 3. Relationbetweenintensitiesofbrightestpeaks From eq.B14 we can also estimate the relation between the intensities of the brightest peaks in the diffraction spec- trum of two FL belonging to the same class. Using the ex- samplewiththeaidofacomputer-controlledXYstageatcon- pression in eq.B14 and assuming k⊥∆ ≈ 0 we can write stantspeedof50µm/s. Thenominalprecisionofthetrans- sin2(k⊥∆)/(k⊥∆)2 −1 ≈ (k⊥∆)2/9. Using the condition lation stage is 0.5µm for the conditions used in this experi- fork⊥ ≈ 0andfollowingacalculationsimilartothattode- ment. A frequency doubled diode pumped Nd:YWO4 solid terminedrelationbetweenthepositionsofthebrightestpeaks state laser (Coherent Verdi V5) emitting a CW beam at 532 wefindthat(k⊥1∆1)2 =(k⊥0∆0)2/τ2. Thereforewehave nmhasbeenusedaslightsourceforDLW.Thebeamwassuit- ablyattenuatedbyaseriesofneutraldensityfiltersandsentto 1 I(η1q,η1)≈I(q,η0)+ τ(1−I(q,η0)), (C5) afocusingmicroscopeobjective(Olympus100X/0.80)sothat thepoweraftertheobjectivewassetat17mW.Thesubstrate meaning that the intensities of brightest peaks of the scaled usedtoengravetheopticalstructuresisaslabofphotorefrac- lattice are more intense of those of the original lattice by a tive lithium niobate doped with iron at the nominal concen- termproportionaltothedifferencebetweenthemaximumat- tration of 0.1 mol% in the melt. The sample was X-cut with tainableintensityandtheintensityoftheoriginallatticeinten- dimensions(X×Y×Z)1mm×8mm×13mmandthelines sities. werewrittenontheXface,byscanningalongtheYdirection Infig.8weplotthequantities(bluedots)τ−1(1−I(q,η0)) with an ordinarily polarized beam. This process can induce and (red cross) (I(η1q,η1) − I(q,η0)) for q such that extraordinaryrefractiveindexchangesaslargeas10−3inthe I(q,η0) > 0.5 and for lattices of N = 300 sites and η0 = written lines and therefore can be used to produce arbitrary 6/11andη1 =1+1/η0respectively. diffraction structures. The diffraction pattern of these struc- tures was measured with the help of a computer-controlled optical diffractometer in which the sample and the detector AppendixD:Experimentalsetup were mounted on two co-axial goniometers that were inde- pendently controlled by a computer [26]. An optical beam TotestexperimentallythediffractionfromFL’s,aseriesof producedbyaHe-Nelaserat632.8nmwithapowerof4mW quasi-periodic diffraction gratings have been prepared using wasexpanded,polarizedalongtheextraordinarydirectionand a photorefractive direct laser writing (DLW) technique [25]. finally transmitted through the sample surface, resulting in a Thistechniqueconsistsinscanningwithafocusedlaserbeam clearlyvisiblediffractionpattern. Thispatternwasmeasured aphotorefractivesample,engravingonitaseriesoflineswith by a Si photodiode and a lock-in amplifier and recorded on amodifiedrefractiveindexwithrespecttotherestofthesam- the computer as a function of the detector and of the sample ple. Thescanningmovementisperformedbytranslatingthe angle. 9 AppendixE:Experimentaldiffractionpatterns form for S(q) describes adequately the peak intensity in the wholerangeofmeasuredvalues(seefig. 9): 1. Structurefactor Inordertocomparetheexperimentaldatawiththetheoret- S(q)=S0e−λq−qq0 (E2) icalcalculationweneedtotakeintoaccountthatourgratings aremadeupofa(quasi)periodicrepetitionofaregionwitha where the parameters S ,λ,q are determined by a least 0 0 modifiedrefractiveindex,∆n(x). Thisleadstothefact,well square fit in the range q ∈ [0.5,3] µm−1 excluding the last knownfromstandarddiffractiontheory,thatthediffractedin- peaks because the corresponding momenta where compara- tensityinreciprocalspaceisproportionaltotheproductoftwo ble with a length scale of the order of the optical waveguide terms: afirstone,S(q)=(cid:12)(cid:12)(cid:82) ∆n(x)eıxqdx(cid:12)(cid:12)2whichdependson width. We also notice that all measured diffraction patterns thedetailedstructureoftherepeatedunitofthegrating(struc- drop almost to zero outside the interval q ∈ [−4,4] µm−1; turefactor)andasecondtermduetolatticegeometry,which this is due to the fact that our lines have a width determined isthetrueobjectofthisstudy: bythelaserwritingopticswhichisnotsmallerthan2µm,so thatourdiffractionpatterncannotprobe|q|>3µm−1. (cid:12) (cid:12)2 IR(q,η)=S(q)N12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)eıxηnq(cid:12)(cid:12)(cid:12)(cid:12) =S(q)I(q,η) (E1) 2. Comparisonwiththeoreticalpatterns n The structure factor modulates the intensity of the lattice In Fig. 10 we compare the theoretical diffraction pattern diffractionpattern,complicatingthecomparisonbetweenex- I(q,η) with the experimental data points for the grating ηb. periments and theory. In principle S(q) could be calculated 3 Theintensityoftheexperimentalpointshasbeenrescaledto byknowingthedetailsoftherefractiveindexprofilechanges take into account the contribution of the structure factor of produced by our technique. Here we used another approach the grating and the q axis of the experimental plot has been whichexploitsthefactoursamplesdifferonlyforthelinepo- rescaledinordertocompareitwiththesimulation,whichcon- sitionsequencex . Wecanusethereforetheperiodicgrating n siders FL’s with S = 1. A similar figure is obtained for the (Fig.9)tomeasurethefunctionS(q)directlyatthereciprocal caseηa =17/6. Theagreementisverysatisfactory: notonly latticepoints{qi }oftheperiodicgrating,whereI(qi )has 1 M M thepositionbutalsotheintensityofthediffractionpeaksare localmaxima. correctlyobtained,confirmingthatourapproachisreliable. I q 10(cid:45)4 I(q,η b) 10(cid:45)6 10(cid:45)8(cid:72) (cid:76) 10(cid:45)10 q q (cid:45)2 0 2 4 Figure 10: (Color online). Experimental diffraction pattern (solid Figure9: (Coloronline). Experimentaldiffractionpatternofthepe- redtopcurve)comparedwiththetheoreticaldiffractionpatternwith riodicgratingwithspacingL=23µm(blue,solidcurve)andfitof theinclusionofthestructurefactor(solidbluebottomcurve)fora thesatellitepeakintensitiesusingfunctionE2(red,dashedcurve). FibonaccigratingwithN = 300linesandL = 23 µmandS = 15µm,ηb =1.875. 3 We found that the following phenomenological functional [1] D.Shechtman,I.Blech,D.Gratias,andJ.W.Cahn,Phys.Rev. [4] C. Janot, Quasicrystals - A primer (second edition), Oxford Lett.53,1951(1984). UniversityPress(1994). [2] D.Levine,andP.J.Steinhardt,Phys.Rev.B34,596(1986). [5] Y. E. Kraus, and O. Zilberberg, Phys. Rev. Lett. 109, 116404 [3] M.Senechal,Quasicrystalsandgeometry,CambridgeUniver- (2012). sityPress(1995). [6] Y.E.Kraus,Y.Lahini,Z.Ringel,M.Verbin,andO.Zilberberg, 10 Phys.Rev.Lett.109,106402(2012). [18] Ch.Li,H.Cheng,R.Chen,Tian.Ma,Li-G.Wang,Yu.Song, [7] M.Verbin,O.Zilberberg,Y.E.Kraus,Y.Lahini,andY.Silber- andHai-Q.Lin,Appl.Phys.Lett.103,172106(2013). berg,Phys.Rev.Lett.110,076403(2013). [19] L.DalNegro(ed.)OpticsofAperiodicStructures: Fundamen- [8] M.Lohse,C.Schweizer,O.Zilberberg,M.Aidelsburger,andI. talsandDeviceApplications,PanStanfordUniversityPublish- Bloch,Nat.Phys.3584(2015). ing(2014). [9] K.Singh,K.Saha,S.A.Parameswaran,andD.M.Weld,Phys. [20] G.Radons,W.JustandP.Hussler(eds.)CollectiveDynamics Rev.A92,063426(2015). ofNonlinearandDisorderedSystems,Springer-Verlag,Berlin [10] R.W.Peng,Y.M.Liu,X.Q.Huang,F.Qiu,MuWang,A.Hu, -Heidelberg(2005). S.S.Jiang, D.Feng, L.Z.Ouyang, andJ.Zou, Phys.Rev.B [21] P.Buczek, L.Sadun, andJ.Wolny, ActaPhysicaPolonicaB, 69,165109(2004). 36,3(2005). [11] A.Lin,Y.K.Zhong,S.M.Fu,C.W.Tseng,andS.L.Yan,Opt. [22] J.-M.LuckPhys.Rev.B39,5834(1989). Expr.22(53)a880(2014). [23] J.-M. Gambaudo, and P. Vignolo, New J. 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Goodwin,arXiv:1508.05909.

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